This seems to be a question about the history of mathematics (a subtle beast, if one is interested in going beyond "Whig history" narratives) - but it is not a question about opinion or argument, in my view: and MO seems a good place for people who know some of these details well to contribute. (See Franz Lemmermeyer's comment below regarding Koenigsberg and Euler.) It's certainly a damn sight better, in my opinion, than the Why Has No One Categorified Rice Pudding? question.
–
Yemon ChoiApr 25 '10 at 0:41

To the original poster: I don't intend to make a habit of it ;) but in a couple of cases I felt that the questions were fine, or at least had attracted worthwhile answers. Anyway, I only have the one vote to re-open; what others do is their choice!
–
Yemon ChoiApr 25 '10 at 9:26

19 Answers
19

Shannon's work on Information theory. Maybe the math wasn't new but the ideas (such as positing a qualitative metric of information and identifying its relevance to design of communication systems) definitely were.

I don't think it's fair to characterize Shannon's information theory as completely new. In fact, the idea of a qualitative matric of information was quite well-known under the name of enthropy in physics (statistical mechanics was developed in 19th-first half of 20th century). Shannon introduced that concept into math and formalized it; but he didn't pretend to invent the idea, and even used the same name!
–
Ilya NikokoshevApr 25 '10 at 6:01

Hermann Weyl wrote in a 1939 article on invariant theory: "The Theory
of Invariants came into existence about the middle of the nineteenth century
somewhat like Minerva: a grown-up virgin, mailed in the shining armor of
algebra, she sprang forth from Cayley's Jovian head. Her Athens over which
she ruled and which she served as a tutelary and beneficent goddess was
projective geometry."

I have often wondered how true this is, as evidenced by my earlier question (mathoverflow.net/questions/124011/…) which I suppose you have seen, since you commented on Noah S's response. Still, I am left wondering how much of Cohen's forcing was "completely new" (not that I can think of a great way to measure such a thing)...
–
Benjamin DickmanMay 25 '14 at 5:23

1

@BenjaminDickman: It's hard to measure anything like that. The analogies that I've seen center on the notion of generic set and how that is similar to other notions of genericity. Even if that idea wasn't Cohen's, his use of genericity is completely new. It's interesting that the various aspects of genericity in all areas of mathematics started coming together into a whole around that time, so there was definitely something in the air...
–
François G. Dorais♦May 25 '14 at 9:47

This happened hundreds of times in physics throughout the twentieth century, because physicists were specifically trained to do mathematics from scratch. The main reason is that it was too time consuming in pre-internet times to learn the specialized jargon of each subfield, so it was easier just to rederive the stuff.

The most significant early success of this sort of willful ignorance is probably the development of special relativity from essentially nothing. The Minkowski geometry of relativity is remarkable, because if you interpret the words "point" and "line" as usual, and the word "circle" as a unit hyperbola with 45-degree angle asymptotes (the unit circle of relativity), it satisfies all the explicit axioms of Euclid's geometry, as set out in the elements, including the axiom of parallels, but is not Euclidean. The essential difference is that circles are not closed curves, so that certain implicit betweenness properties fail. There are distinct points which are at a zero "distance" from one another, the hypotenuse of a right triangle is always shorter than one of the sides, etc. This is amazing to me, because of the number of people who have considered models of geometry before Einstein (including all the heavy focus on non-Euclidean geometry for the previous century). All the bigwigs missed Minkowski geometry.

Aside from Einstein's work, there are the following mathematical developments from physics, all of which came out of nowhere mathematically:

Quantum mechanics, in particular, the theory of the canonical commutation relation [x,p]=i and its relationship with wave operators and random walks.

Dirac's distribution theory (delta-functions): this completed the notion of Eigenvalue of a linear operator to include Eigenvalues and Eigenfunctions for the x operator in quantum mechanics.

Majorana spinors--- these were due to the discovery of the Dirac equation. The representation theory of SO(p,q) is now entirely dependent on dirac matrices and the Majorana and Weyl conditions.

Wigner's random matrix theory. This was completely ab-initio, and is now very active mathematics.

Anderson localization: this is also a mathematical surprise--- the eigenfunctions of randomized potentials are localized in space. The full resulting theory has still not been made part of mathematics, but Anderson's paper is an ab-initio (although not rigorous) argument.

Metropolis algorithm--- this essentially inaugurated monte-carlo methods, and I do not know any previous work it builds on.

Feynman's path integral--- this was developed within mathematics as the Wiener integral at about the same time, but the physics work is completely ab-initio. Needless to say, the results are not going into mathematics easily (in my opinion, this is mostly due to the reluctance of mathematicians to make every subset of R measurable).

Candlin's fermionic path integral (Berezin integrals)--- Candlin in 1956 develops the whole theory of path integrals for fermionic fields from scratch in a Neuvo Cimento article with next to no citations (in either direction). The theory was ignored for a decade for no apparent reason.

Kraichnan's inverse cascade--- generally the statistical theory of nonlinear classical equations is developed from scratch by Kraichnan and others. The biggest shocker is the inverse cascade--- in two dimensions, eddies go up from small scales to big scales.

Zimmermann's forest formula--- this is now part of mathematics, due to Kreimer and Connes, but Zimmermann did it from scratch in physics.

The theory of second order phase transitions and modern renormalization by Widom/Wilson.

Wilson's theory of operator product expansions, (which is not a part of mathematics yet)

Supersymmetry is developed from scratch by several groups with no previous motivation in mathematics (not much in physics). The original germ of an idea is in Golfond and Likhtman, but the person who does most of the early theory work is Pierre Ramond. Wess and Zumino's work also comes out of nowhere.

Virasoro algebra/Kac-Moody algebra-- Virasoro algbera is the theory of infinitesimal conformal maps under composition, so it should have been classical mathematics, but as far as I know, it wasn't. The theory started (as far as I know) with the study of string theory in the early 1970s.

Mirror symmetry--- this owes to previous work in T-duality in string theory, not in mathematics.

Witten's global anomalies--- these are not yet part of rigorous mathematics, but they are ab-initio, and were a complete surprise.

I got tired, but there are hundreds, maybe thousands of examples, because all the results in the physics literature were generally ab-initio. It is a standard practice for some mathematicians to scan the physics literature for original ideas and incorporate them into mathematics.

The "owe little or nothing to previous work" part of the question seems to disqualify most answers from physics, since such concepts often build on earlier work on physical problems. For example, Einstein's formulation of special relativity owes a big debt to Maxwell's work on electromagnetism (Minkowski's discovery of spacetime geometry was inspired by Einstein's paper), and Dirac's distributions are derived from Heaviside's. The history of the Virasoro algebra dates to 1909 and is covered in brief in the Wikipedia page.
–
S. Carnahan♦Aug 1 '11 at 7:43

1

Fair enough--- but the usual way mathematics is done is by quoting and using previously proven theorems, and the mathematical work of the physicists generally does not quote previous theorems, but instead constructs the objects in question from scratch. So I think it is in the spirit of the question. The Dirac and Virasoro examples might be inappropriate, I don't know the history of the things very well.
–
Ron MaimonAug 2 '11 at 17:42

1

---the mathematical work of the physicists generally does not quote previous theorems--- Well, take "random matrix theory" for instance. What exactly was new there: the notion of the random variable, the matrix, the spectrum, or the method of moments? My opinion is that physicists (as well as engineers/biologists/...) can get credit for asking a multitude of questions no mathematician would ask otherwise, and here I take my hat off. Beyond that, they just use whatever tools are already there and if that is not enough, just engage in educated guesses and wishful thinking.
–
fedjaMay 24 '14 at 22:06

Quite a few (maybe most?) of your examples have antecedents in mathematics - relativity (Poincare), spin representations (Chevalley), etc. - and in many cases it is not as clear as you claim that physicists were completely unaware of the mathematics. Physicists deserve a lot of credit for implicitly suggesting interesting mathematical problems, but I think you overstate the extent to which they actually invent new mathematics.
–
Paul SiegelMay 25 '14 at 4:28

The solution of the cubic equation by Scipione del Ferro and Tartaglia
in the early 16th century. This was not only a great advance in algebra,
but it also forced mathematicians to confront complex numbers.

The idea that the cubic equation can be "solved" surely owes a debt to the notion that the quadratic equation can be solved...
–
Qiaochu YuanApr 18 '10 at 17:38

2

True, but since it took thousands of years to get beyond the solution of the quadratic, I think that something extra was involved.
–
John StillwellApr 18 '10 at 22:56

7

Feynman mentions this example in one of his books (I think What Do You Care What Other People Think?!) as an important realization to people living at the time, that they could do something that the ancient Greeks could not.
–
Todd Trimble♦Jun 12 '11 at 12:23

Can you qualify that? That $\sqrt{2}$ is irrational has been known for a long time, and that it 'exists' is 'clear' from simple geometrical constructions. I am not saying you are wrong, but I really think your answer needs expanded!
–
Jacques CaretteApr 16 '10 at 14:49

7

It may have been known for a long time, but somebody had to discover it! Perhaps Hippasus of Metapontum, about 2500 years ago. It must have been as unexpected as Cantor's infinities.
–
TonyKApr 16 '10 at 16:26

I am not sure what exactly do you mean by this. In the early years, Ramanujan discovered lots of interesting and important formulas, and later proved them and some theorems, but originally these were not "concepts". Later on in his life he did introduced some concepts (notably en.wikipedia.org/wiki/Mock_theta_function ) but they were clearly related to some earlier work.
–
Igor PakApr 16 '10 at 17:58

Fair point Igor. I was simply emphasizing Ramanujan's isolationist nature. Since he was almost completely unaware of earlier work, one could reasonably say that he could not owe a debt to it. But you are right that he did not introduce completely novel concepts.
–
Tony HuynhApr 16 '10 at 19:17

5

The idea that Ramanujan came up with math from nowhere is an urban legend: It is a fun idea, so it is passed on without being checked. Some urban legends are true, and some are not. I'd like to see references.
–
Douglas ZareApr 16 '10 at 22:52

13

I suppose that the mathematics professor from Good Will Hunting doesn't count as a legitimate reference?
–
Tony HuynhApr 17 '10 at 1:08

It's difficult to be certain with Ramanujan - most of his methods are completely unknown.
–
teilApr 18 '10 at 13:08

I strongly disagree with the assessment of Frege; plenty of others helped pave the way, including for example Boole. I am a little skeptical of surreal numbers as well.
–
Todd Trimble♦Jun 12 '11 at 12:27

1

@Todd: have you read Boole's actual work on logic, and compared it to what Frege wrote? [I have recently read a number of papers by both.] They are really quite different. Frege's work is infused with a lot of philosophy and deep 'foundational' thinking about all of mathematics. Boole's work is fantastic, but in a different direction.
–
Jacques CaretteJun 13 '11 at 2:33

4

What I had in mind when I wrote that was that Boole and others paved the way for the realization that logic could be mathematicized. My understanding is that Boole's work shows how propositional logic can be represented in symbolic, algebraic form. Subsequently, others like E. Schroeder and C.S. Peirce had pushed the algebraization of relational calculus quite far (including of course relational composition, closely tied to quantification). Frege in fact knew of this work but was somewhat dismissive. Anyway, pursuit of the analogies between algebra and FOL was quite vigorous before Frege.
–
Todd Trimble♦Jul 31 '11 at 0:04

Écalle's work on resummation and resurgent functions. While there is a bit of work that pre-dates him, the vast bulk of his theory is really novel and built 'from scratch'. This is especially clear to anyone who has ever tried to read the Orsay preprints of his original manuscripts on resurgent functions! [The only notation more spectacular than his was Frege's]

There were no graphs in Euler's solution; the translation of Euler's idea into graph theory came later. See "The truth about Koenigsberg" in "Leonhard Euler. Life, work and legacy" (Bradley and Sandifer, eds.).
–
Franz LemmermeyerApr 16 '10 at 13:47

Stallings's bipolar structures created to prove that groups of cohomological dimension 1 are free.

(Stallings might not have agreed with my nomination, but his statement at the end of the paper that his techniques are a result of "meditating on the proof of the Sphere Theorem" somehow makes his work even more remarkable to me.)

It seems like Dirichlet's Theorem on Primes in Arithmetic Progressions came out of nowhere, or at least his methods of proof. While the complex analysis may not have been new, his application of it, through the Dirichlet characters and the series he made from them, to number theory was pretty novel.

Pcf theory/cardinal arithmetic. Well, it's not exactly built from scratch, but there are plenty of nice results which do not use any sophisticated metamathematical machinery (such as forcing, inner models, etc).

Edit: I've deleted part of my answer due to a little misunderstanding.

Point-set topology owes a great deal to whatever was known about metric spaces at the time. I don't think you can reasonably claim that the concept doesn't owe a dept to previous work.
–
Qiaochu YuanApr 18 '10 at 17:37

Of course, you're right. It seems that I misread the original question. I've now deleted the bad part.
–
HaimApr 18 '10 at 17:46

1

A big chunk of Shelah's work, in general, seems to have come out of nowhere!
–
David FernandezBretonApr 5 '12 at 6:29

Category theory must be here too - although it was created not so much as something out of the blue but rather to organise and interrelate the accumulated body of mathematical knowledge (according to Eilenberg and MacLane categories were invented to formulate rigorously the intuitive notion of natural transformation), still I think it was a completely new approach to the very idea of abstraction in mathematics which I believe has yet to show us its full potential.